An Improved Nyquist–Shannon Irregular Sampling Theorem From Local Averages

The Nyquist–Shannon sampling theorem is on the reconstruction of a band-limited signal from its uniformly sampled samples. The higher the signal bandwidth gets, the more challenging the uniform sampling may become. To deal with this problem, signal reconstruction from local averages has been studied in the literature. In this paper, we obtain an improved Nyquist–Shannon sampling theorem from general local averages. In practice, the measurement apparatus gives a weighted average over an asymmetrical interval. As a special case, for local averages from symmetrical interval, we show that the sampling rate is much lower than that of a result by Gröchenig. Moreover, we obtain two exact dual frames from local averages, one of which improves a result by Sun and Zhou. At the end of this paper, as an example application of local average sampling, we consider a reconstruction algorithm: the piecewise linear approximations.

[1]  Xingwei Zhou,et al.  Approximation of signals from local averages , 2006, Appl. Math. Lett..

[2]  Holger Boche,et al.  Behavior of the Quantization Operator for Bandlimited, Nonoversampled Signals , 2010, IEEE Transactions on Information Theory.

[3]  N. Levinson.  Gap and Density Theorems , 1940 .

[4]  Holger Boche,et al.  Behavior of Shannon’s Sampling Series for Hardy Spaces , 2008 .

[5]  Paul L. Butzer,et al.  Errors in truncated sampling series with measured sampled values for not-necessarily bandlimited functions , 1998 .

[6]  Hans G. Feichtinger,et al.  Theory and practice of irregular sampling , 2021, Wavelets.

[7]  A. J. Jerri.  Correction to "The Shannon sampling theorem—Its various extensions and applications: A tutorial review" , 1979 .

[8]  O. Christensen.  An introduction to frames and Riesz bases , 2002 .

[9]  K. Gröchenig.  RECONSTRUCTION ALGORITHMS IN IRREGULAR SAMPLING , 1992 .

[10]  Holger Boche,et al.  On the behavior of Shannon's sampling series for bounded signals with applications , 2008, Signal Process..

[11]  A. J. Jerri.  The Shannon sampling theorem—Its various extensions and applications: A tutorial review , 1977, Proceedings of the IEEE.

[12]  Weidong Chen.  A Fast Convergence Algorithm for Band-Limited Extrapolation by Sampling , 2009, IEEE Transactions on Signal Processing.

[13]  F. Marvasti.  Nonuniform sampling : theory and practice , 2001 .

[14]  Wenchang Sun,et al.  Approximation of weak sense stationary stochastic processes from local averages , 2007 .

[15]  Wenchang Sun,et al.  Reconstruction of Band-Limited Functions from Local Averages , 2002 .

[16]  Wenchang Sun,et al.  An Average Sampling Theorem for Bandlimited Stochastic Processes , 2007, IEEE Transactions on Information Theory.

[17]  R. Young,et al.  An introduction to nonharmonic Fourier series , 1980 .

[18]  M. Unser.  Sampling-50 years after Shannon , 2000, Proceedings of the IEEE.

[19]  K. Gröchenig.  Irregular sampling of wavelet and short-time Fourier transforms , 1993 .

[20]  Wenchang Sun,et al.  Reconstruction of band-limited signals from local averages , 2002, IEEE Trans. Inf. Theory.

[21]  P. P. Vaidyanathan,et al.  Generalized sampling theorems in multiresolution subspaces , 1997, IEEE Trans. Signal Process..

[22]  A. Aldroubi.  Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces , 2002 .

[23]  R. Wiley,et al.  Recovery of Bandlimited Signals from Unequally Spaced Samples , 1978, IEEE Trans. Commun..

[24]  J. R. Higgins,et al.  Five short stories about the cardinal series , 1985 .

[25]  Xiang-Gen Xia,et al.  A method with error estimates for band-limited signal extrapolation from inaccurate data , 1997 .

[26]  H. Schwarzlander,et al.  Demodulation Procedure for Very Wide-Band FM , 1977, IEEE Trans. Commun..